Hello Future Calculus Level One Student,
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1 Hello Future Calculus Level One Student, This assignment must be completed and handed in on the first day of class. This assignment will serve as the main review for a test on this material. The test will be administered during the first week of classes. You must complete the entire assignment showing sufficient evidence of effort in mathematical reasoning, use of computational skills, understanding of concepts, and communication of appropriate mathematical processes and terms. In other words, show all your work or explain how you arrived at the solution, circle your answers, label when necessary, and answer each word problem in complete sentences, etc. You should only use pencil to complete these problems. No pens, markers, etc. allowed. Calculators may be used and are recommended. You will be expected to have a TI-8 or TI-84. When using calculators, show work by writing any expression that you enter into the calculator. If you have any questions, feel free to whitehilln@region10ct.org Have a great summer! I look forward to seeing you all next year! Good luck and enjoy, Ms. Whitehill Calculus Summer Homework 1
2 Summer Review Packet for Students Entering Calculus Complex Fractions When simplifying complex fractions, multiply by a fraction equal to 1, which has a numerator and denominator composed of the common denominator of all the denominators in the complex fraction. Example: 7 6 x x + 1 Example: 2 x + x x x 4 = 7 6 x + 1 x x + 1 = 7x 7 6 7x 1 = 5 5 x + 1 = 2 x + x x x 4 x(x 4) 2(x 4) + x(x) 2x x2 = = x(x 4) 5(x)(x 4) 1(x) 5x 2 20x x = x2 2x + 8 5x 2 21x Simplify each of the following. 1) 25 a - a 5 + a 2) 2-4 x x + 2 ) x x - Functions To evaluate a function for a given value, simply plug the value into the function for x. Recall: ( f g)(x) = f (g(x)) OR f [g(x)] read f of g of x means to substitute the inside function (in this case g(x) ) in for x in the outside function (in this case, f(x)). Example: Given f (x) =2x 2 +1 and g(x) = x - 4 find f(g(x)). f (g(x)) = f (x - 4) = 2(x - 4) = 2(x 2-8x + 16) + 1 = 2x 2-16x f (g(x)) = 2x 2-16x + Calculus Summer Homework 2
3 Let f (x) = 2x +1 and g(x) = 2x 2-1. Find each. 6) f (2) = 7) g(-) = 8) f (a +1) = 9) f éë g(-2) ù û = 10) g éë f (m + 2) ù û = 11) f (x + h) - f (x) h = Let f (x) = sin x Find each exactly. æ p ö æ 2p ö 12) f è ç 2 ø = 1) f è ç ø = Let f (x) = x 2, g(x) = 2x + 5, and h(x) = x 2-1. Find each. 14) héë f (-2) ù û = 15) f éë g(x -1) ù û = 16) g é ë h(x ) ù û = f (x + h) - f (x) Find for the given function f. h 17) f (x) = 9x + 18) f (x) = 5-2x Calculus Summer Homework
4 Intercepts and Points of Intersection To find the x-intercepts, let y = 0 in your equation and solve. To find the y-intercepts, let x = 0 in your equation and solve. Example: y = x 2-2x - x - int. (Let y = 0) 0 = x 2-2x - 0 = (x - )(x + 1) x = -1 or x = x - intercepts (-1,0) and (,0) y - int. (Let x = 0) y = 0 2-2(0) - y = - y - intercept (0,-) Find the x and y intercepts for each. 19) y = 2x ) y = x 2 + x ) y = x 16 - x 2 22) y 2 = x - 4x Use substitution or elimination method to solve the system of equations. Example: x 2 + y - 16x + 9 = 0 x 2 - y 2-9 = 0 Elimination Method 2x 2-16x + 0 = 0 x 2-8x + 15 = 0 (x - )(x - 5) = 0 x = and x = 5 Plug x= and x = 5 into one original 2 - y 2-9 = y 2-9 = 0 -y 2 = 0 16 = y 2 y = 0 y = ±4 Points of Intersection (5, 4), (5,-4) and (, 0) Calculus Summer Homework 4 Substitution Method Solve one equation for one variable. y 2 = -x x - 9 (1st equation solved for y) x 2 - (-x x - 9) - 9 = 0 Plug what y 2 is equal to into second equation. 2x 2-16x + 0 = 0 (The rest is the same as x 2-8x + 15 = 0 previous example) (x - )(x - 5) = 0 x = or x - 5
5 Find the point(s) of intersection of the graphs for the given equations. 2) x + y = 8 24) x2 + y = 6 25) x2-4y 2-20x - 64y = 0 4x - y = 7 x + y = 4 16x 2 + 4y 2-20x + 64y = 0 Interval Notation 26) Complete the table with the appropriate notation or graph. Solution Interval Notation Graph -2 < x 4-1,7) éë Solve each equation. State your answer in BOTH interval notation and graphically. 27) 2x -1 ³ 0 28) -4 2x - < 4 29) x 2 - x > 5 8 Calculus Summer Homework 5
6 Domain and Range Find the domain and range of each function. Write your answer in INTERVAL notation. 0) f (x) = x 2-5 1) f (x) = - x + 2) f (x) = sin x ) f (x) = 2 x - 1 Inverses To find the inverse of a function, simply switch the x and the y and solve for the new y value. Example: f (x) = x + 1 Rewrite f(x) as y y = x + 1 Switch x and y x = y + 1 Solve for your new y ( x) = y + 1 x = y + 1 y = x - 1 ( ) Cube both sides f -1 (x) = x - 1 Find the inverse for each function. Simplify Solve for y Rewrite in inverse notation 4) f (x) = 2x +1 5) f (x) = x2 Calculus Summer Homework 6
7 Also, recall that to PROVE one function is an inverse of another function, you need to show that: f (g(x)) = g( f (x)) = x Example: If: f (x) = x - 9 and g(x) = 4x + 9 show f(x) and g(x) are inverses of each other. 4 æ f (g(x)) = 4 x - 9 ö è ç 4 ø + 9 g( f (x)) = ( 4x + 9) = x = 4x = x = 4x 4 = x f (g(x)) = g( f (x)) = x therefore they are inverses of each other. Prove f and g are inverses of each other. 6) f (x) = x 2 g(x) = 2x 7) f (x) = 9 - x 2, x ³ 0 g(x) = 9 - x Equation of a line Slope intercept form: y = mx + b Vertical line: x = c (slope is undefined) Point-slope form: y - y 1 = m(x - x 1 ) Horizontal line: y = c (slope is 0) 8) Use slope-intercept form to find the equation of the line having a slope of and a y-intercept of 5. 9) Determine the equation of a line passing through the point (5, -) with an undefined slope. Calculus Summer Homework 7
8 40) Determine the equation of a line passing through the point (-4, 2) with a slope of 0. 41) Use point-slope form to find the equation of the line passing through the point (0, 5) with a slope of 2/. 42) Find the equation of a line passing through the point (2, 8) and parallel to the line y = 5 6 x ) Find the equation of a line perpendicular to the y- axis passing through the point (4, 7). 44) Find the equation of a line passing through the points (-, 6) and (1, 2). 45) Find the equation of a line with an x-intercept (2, 0) and a y-intercept (0, ). Calculus Summer Homework 8
9 Radian and Degree Measure 180 Use to get rid of radians and p radians convert to degrees. 46) Convert to degrees: a. 5p 6 b. p radians Use to get rid of degrees and 180 convert to radians. 4p c. 2.6 radians 5 47) Convert to radians: a. 45 b. -17 c. 27 Angles in Standard Position 48) Sketch the angle in standard position. a. 11p b. 20 c. - 5p 6 d. 1.8 radians Reference Triangles 49) Sketch the angle in standard position. Draw the reference triangle and label the sides, if possible. a. 2 p b. 225 Calculus Summer Homework 9
10 Unit Circle You can determine the sine or cosine of a quadrantal angle by using the unit circle. The x-coordinate of the circle is the cosine and the y-coordinate is the sine of the angle. Example: sin90 = 1 cos p 2 = 0 50) a.) sin180 b.) cos270 c.) sin(-90 ) d.) sinp e.) cos 60 f.) cos(-p) Graphing Trig Functions y = sin x and y = cos x have a period of 2p and an amplitude of 1. Use the parent graphs above to help you sketch a graph of the functions below. For f (x) = Asin(Bx + C) + K, A = amplitude, 2p B = period, C = phase shift (positive C/B shift left, negative C/B shift right) and K = vertical shift. B Graph two complete periods of the function. 51) f (x) = 5sin x 52) f (x) = sin2x æ 5) f (x) = - cos x - p ö è ç 4 ø 54) f (x) = cos x - Calculus Summer Homework 10
11 Trigonometric Equations: Solve each of the equations for 0 x < 2p. Isolate the variable, sketch a reference triangle, find all the solutions within the given domain, 0 x < 2p. Remember to double the domain when solving for a double angle. Use trig identities, if needed, to rewrite the trig functions. (See formula sheet at the end of the packet.) 55) sin x = ) 2cos x = 57) SKIP 58) sin 2 x = ) sin2x = ) 2cos 2 x -1- cos x = 0 61) SKIP 62) sin 2 x + cos2x - cos x = 0 Calculus Summer Homework 11
12 Inverse Trigonometric Functions: Recall: Inverse Trig Functions can be written in one of ways: arcsin ( x) sin -1 ( x) Inverse trig functions are defined only in the quadrants as indicated below due to their restricted domains. cos -1 x sin -1 x cos -1 x tan -1 x sin -1 x tan -1 x Example: Express the value of y in radians. y = arctan -1 Draw a reference triangle. 2-1 This means the reference angle is 0 or p 6. So, y = p 6 so that it falls in the interval from -p 2 < y < p 2 Answer: y = p 6 For each of the following, express the value for y in radians. 6) y = arcsin ) y = arccos( -1) 65) y = arctan(-1) Calculus Summer Homework 12
13 Example: Find the value without a calculator. æ cos arctan 5 ö è ç 6ø 61 Draw the reference triangle in the correct quadrant first. 5 Find the missing side using Pythagorean Thm. 6 Find the ratio of the cosine of the reference triangle. cosq = 6 61 For each of the following give the value without a calculator. æ 66) tan arccos 2 ö æ 12ö è ç ø 67) sec sin -1 è ç 1ø 68) SKIP 69) SKIP Vertical Asymptotes Determine the vertical asymptotes for the function. Set the denominator equal to zero to find the x-value for which the function is undefined. That will be the vertical asymptote. 74) f (x) = 1 x 75) f (x) = x 2 76) f (x) = 2 + x 2 x 2-4 x 2 (1- x) Horizontal Asymptotes Determine the horizontal asymptotes using the three cases below. Case I. Degree of the numerator is less than the degree of the denominator. The asymptote is y = 0. Case II. Degree of the numerator is the same as the degree of the denominator. The asymptote is the ratio of the lead coefficients. Case III. Degree of the numerator is greater than the degree of the denominator. There is no horizontal asymptote. The function increases without bound. (If the degree of the numerator is exactly 1 more than the degree of the denominator, then there exists a slant asymptote, which is determined by long division.) Calculus Summer Homework 1
14 Determine all Horizontal Asymptotes. 77) f (x) = x2-2x + 1 x + x ) f (x) = 5x - 2x x - x ) f (x) = 4x5 x 2-7 Logarithms and Exponentials Recall: There are laws of exponents and laws of logarithms that can be used to solve logarithmic and exponential equations. ln x = log e x where e is an irrational number approximately equal to log x = log 10 x (useful for calculator) Example Solve the following logarithmic equation. log log 5 2x 2 = log 5 48 Use product rule to combine logs: log 5 6(2x 2 ) = log 5 48 log 5 12x 2 = log 5 48 Since logs and exponentials are inverse operations, put the log base as the exponential base of each side to cancel the logs. 5 log 5 12x 2 = 5 log x 2 = 48 Solve for x. x 2 = 4 x = ±2 Example Solve the following exponential equation. 6e 5x 6 4 = 50 Isolate the exponential piece: 6e 5x 6 = 54 e 5x 6 = 9 Take the log or ln of each side and use power rule to get the exponent in front of the logarithm. ln e 5x 6 = ln 9 (5x 6) ln e = ln 9 5x 6 = ln 9 (since ln e = 1) 5x 6 = 2.2 Solve for x. 5x = 8.2 x = ) 8 5a 5 = 5 81) 2 8 4r+1 6 = 0 82) ln(4x 1) ln = 5 8) log 8 (n ) = log 8 ( 9n 2) Calculus Summer Homework 14
15 Reciprocal Identities: csc x = 1 sin x Formula Sheet sec x = 1 cos x cot x = 1 tan x Quotient Identities: tan x = sin x cos x cot x = cos x sin x Pythagorean Identities: sin 2 x + cos 2 x = 1 tan 2 x +1 = sec 2 x 1+ cot 2 x = csc 2 x Double Angle Identities: Logarithms: sin2x = 2sin xcos x tan2x = 2 tan x 1- tan 2 x y = log a x is equivalent to x = a y Product property: log b mn = log b m + log b n cos2x = cos 2 x - sin 2 x = 1-2sin 2 x = 2cos 2 x - 1 Power property: log b m p = plog b m Quotient property: log b m n = log b m - log b n Property of equality: If log b m = log b n, then m = n Change of base formula: log a n = log b n log b a Slope-intercept form: y = mx + b Point-slope form: y - y 1 = m(x - x 1 ) Standard form: Ax + By + C = 0 Calculus Summer Homework 15
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